= Solution
Initialize two <qubits> in $|00\rangle$. A <Hadamard gate> on the first followed by a <controlled-NOT gate> $1\to2$ prepares the <Bell state> $|\phi_+\rangle$. The permitted <T gates> and <Hadamard gates> implement the needed <Pauli gates> exactly:
$$
Z=T^4,\qquad X=HT^4H.
$$
Apply $T_1^4$, followed by $H_2,T_2,T_2,T_2,T_2,H_2$. These are a <Pauli Z gate> on the first <qubit> and a <Pauli X gate> on the second, giving
$$
\frac{|00\rangle+|11\rangle}{\sqrt2}
\xrightarrow{Z_1}\frac{|00\rangle-|11\rangle}{\sqrt2}
\xrightarrow{X_2}\boxed{\frac{|01\rangle-|10\rangle}{\sqrt2}.}
$$
Every operation is in the allowed gate set; no <quantum measurement> or probabilistic branch is needed.
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